 The next object we are going to study is called quotient spaces, all right. And there is a reason why students often have a sort of a phobia about this more than any other kind of you know operations that you do on vector spaces. The reason probably is because we try to understand or gain some familiarity with this in terms of objects that we already know. The first and the most important thing that you would do well to remember about these objects we are about to study that is quotient spaces, they are not objects that should be seen as something similar to objects in the individual vector spaces. You see when we talked about products, it is true that it was a bigger list, but you would not be faulted too much if you thought that this v1 is after all something that belongs in the vector space v1, v2 is something that comes from vector space v2. So these objects taken isolated in isolation at a time, these are objects after all that look exactly like the parent vector spaces. So we have an idea of what they look like, it is just a list, we just padding them along and stacking them up together, right, but when you go into quotient spaces or factor spaces as they are sometimes referred to, right, you have to get out of that mindset and try and understand what these objects are, what they look like, all right. Once you understand that clearly, a lot of the results will be very obvious because again the intuitions that you have about Euclidean spaces will help you, right. So with that sort of a preface, let us launch into this. So as usual, we have a vector space v and we have a subspace of the vector space which is u, right. Just go by the definitions from the very first step onwards and ask me if you have any doubts about any of the things that I am introducing, all right. So the first thing I am going to define is this, okay, already a very weird looking notation and something that we generally say you shouldn't do which is adding apples and oranges, but for want of a better notation, this is how we will represent this. This is not the common addition. This is a subspace of the vector space v, all right and this is a vector v sitting inside the vector space, big v here, all right. So what is basically being talked about here? What sort of a thing is this? First of all, is this a vector space unless v is 0, of course, because this is basically shifted versions of that subspace u, we can belong to you, we will get into that. But if it is in general not so, then this is not a vector space. What is this at best? It is a set exactly as it looks. In fact, it is called an affine set, all right. So for v belonging to the vector space v, we define v plus u in such a manner. If I just try to draw a sketch for Euclidean spaces, let's take R2 for instance is x and y and let's say my subspace is this blue line that is u, all right. And let's say this is v. So what is this set that I have just described to you going to look like? What do you think this set is going to look like here, parallel, absolutely. So okay my sketching is not too good, but yep that's what it is. So this is v plus u. So you see it's a shifted version. The first observation very importantly is there is no one unique v that allows you to define this purple line, right. So you might have chosen this v and your friend might have chosen this v hat and your friend would say hey this is not v plus u, this is v hat plus u and you could argue with your friend till Ragnarok without any conclusion. That is the sort of thing we want to prevent in mathematics, ambiguities, right. So what is it that is sort of missing in all of this? We need some structure with these objects that we are now defining because now we want to carry out some algebra with these objects, right. So this is just one set, yeah. But remember the goal is that I am going to define v, I am going to define u and then I am going to look for all possible parallel translations of u which is the subspace. This is just one parallel translation and in that one parallel translation itself you see there is quite a lot of bickering going on between v and v hat. Like you might say v is the best way to do it, so friends would say no v hat is the best way to do it. Now imagine what we are trying to deal with here is an object defined like so which is now a collection of what exactly v plus u for all possible v's that you choose from the vector space v. And this is the object we are interested in. Now if this is the object that you are interested in then what I have just described in the previous line is just a member of this, is that clear? See the overall object is this, it's a notation that's why it's called a quotient, it's a factor, it's v quotiented by u where u is a subspace of v. So if you take one particular parallel translate or a fine set or sometimes also called coset that is just this object. But what I am interested in looking at is the choice of all possible such parallel translates or cosets that are parallel to u. What is it that is so interesting and why should it interest us? Why should we be bothered with this object? Just to give you some intuition once again I will use some figures, I will draw the 2D and 3D. So suppose this is the x, y, z alright and again let's say this is u passing through the origin alright just assume it's passing through the origin I mean I can't do anything better than that other than to ask you to assume. So this is u. Now very very important suppose this u is not exactly you know just parallel to the y axis in some sense so that every time you translate this u what are you going to get? I am just going to ask you to try and imagine this scenario. You have the x, y, z plane space sorry alright and you have a plane passing through the origin and now you are translating that plane up and down or in you know any possible direction it's basically a translation. So every possible plane will cut the y axis at some point or the other. The point where the y axis perforates the parallel translate of u is a unique point right. Now if you keep doing that of course the subspace is the one that is passing through the origin but any other non subspace coset yeah they will just cut at different different points. So do you immediately realize something interesting going on? How many such parallel translates can you draw? They are exactly as many yeah infinite for sure but they are exactly corresponding to the number of points on the y axis and in fact y y axis choose any other line that perforates this plane and along that line every point corresponds to a corresponds to a fellow such as this. So this entire set is as numerous exactly containing as many members as the number of points in a straight line here's how I would like to set your thought process in motion. This u when it passes through the origin is a subspace of dimension 2 the entire ambient dimension is 3 and now if you think a little deeply about this it turns out that this fellow if it has a vector space structure we don't know that yet we have to define addition and multiplication yet but if it has a vector space structure it seems like it is one dimensional is it not? Because you see for every such parallel translate you are exactly come up with one point. So there seems to be a one to one on two correspondence with a particular straight line that is one dimensional. Now think about the second example which is again 3d let's say this time this is x this is y and this is z and now let's say this is my straight line and this is my u. So a straight line through the origin is also a subspace right. So you think a parallel translates of the straight line they are not planes they are also going to be individually looking like the straight line after all. All parallel translates of a particular object look like the object itself. So what are basically all these other parallel translates of this that is objects like these they are just you know other straight lines shifted from the origin but parallel to this straight line. So each of those parallel lines perforates a particular point on the x y axis sorry x y plane. So basically every point on the x y plane sort of corresponds to a unique object in this set which means again the theme sort of fits you have the three dimensional vector space three dimensional Euclidean space in which you have a one dimensional subspace and how numerous are the parallel translates of that one dimensional subspace you have exactly a one to one correspondence with the two dimensional subspace that is x y plane or any plane for that matter the x y plane I have again chosen for convenience just like I chose the y axis there but the point is if you are if you are already imagining this picture it seems like if you know that the overall parent spaces dimension is something and if you know that this subspace's dimension is something it seems that the dimension of this assuming it is a vector space we have not done that yet I am just trying to give you some intuition assuming this is a dimension vector space its dimension seems to be like the difference between these two seems to intuitively fall in place right but again we have a few miles to go before that we have to at least establish whether this is a vector space what is the addition if it is a vector space and so on and so forth so we have to explore certain properties but just keep this intuition in mind we will revisit this and formally prove this not just for Euclidean spaces but for general vector spaces very shortly we will do that sorry well you look at this so every point on the x y plane x y plane is two dimensional so every point on the x y plane corresponds to one member in this set so that means this set has as many objects as the number of objects in a two dimensional vector space so it should be at least as numerous there must be a isomorphism at least we expect there will be an isomorphism with a two dimensional vector space and this is a one dimensional vector space the ambient is three so three minus one two right so we can expect it to I mean it is a good guess is what I am saying again lot of things to be done here this is a very semi-formal way of approaching this as of now right yes yeah yeah yeah you have to start with a subspace u is always a subspace so these are parallel translates of a subspace which are these affine sets or these cosets right so now we will prove a very important property of these cosets or affine sets we will see that if you know this argument that we sort of motivated towards the beginning like two people coming out with two different vectors v and they eventually merge and collapse onto the same affine set so who is the better one right so it turns out it will matter not as to which way you are describing this either description goes you can as well make peace with your friend so that is the message so what is the point the point is the equivalence of these three conditions the fact that v minus w belongs to you okay let me just write that down here the following are equivalent what are these then v plus u is equal to w plus u and third v plus u intersection w plus u is not an empty set which if you interpret a little closely suggests that two affine sets are either parallel to one another or they are exactly the same as one another there is nothing else you cannot have two affine sets that have something in common but not everything in common you cannot have partial overlap either they are completely non-touching and disjoint if they are not disjoint then they must be the same right that is what this claim would apparently imply yeah I forgot to put the bracket thank you all right so how do we go about proving this okay so here's how we will go about this proof when you have these equivalent sort of proofs sometimes these are handy ways of doing it show the equivalence of instead of showing each one's equivalence to each of the other two show that this implies this this implies this and this implies this and you're done yeah it's a cycle of reasoning all right so first suppose v minus w belongs to you all right now let an object look we already know that this is a set yeah it's a set containing vectors inside the vector space v yeah so suppose v minus w is sitting inside let v hat belong to v plus u implies there exists u1 in u such that v hat is equal to u1 plus v clear that's the definition if something belongs to this objects like this it means that there is some object inside the subspace u which when added to this vector v results in the vector inside that set right but now what can we say v minus w belongs to you so what am I going to replace this with what do you think this is going to be can I not write this as u1 plus v minus w plus w why am I writing it in this weird fashion because I have to use whatever's been given to me and what's been given to me is the fact that this belongs to you so this entire object let me use a different color so this entire object then belongs to what you because this object belongs to you by my proposition here and this object obviously belongs to you as I have assumed it to so therefore this is some u hat plus w sorry so this is some u hat plus w where of course this u hat belongs to you so then this obviously belongs to what this belongs to right w plus u I started with something that belongs in v plus u and I notice that it must belong to w plus u therefore v plus u must be contained inside w plus u right can I go about it the other way anything prevents me from doing that nothing whatsoever yeah I mean just start with an object that belongs to excuse me w plus u therefore it is some w plus some u2 right and then at some point look this is after all a subspace so if this belongs to you then w minus v also belongs to you because you is closed under addition and the additive inverse of any vector inside you must also be in you right so this means basically w minus v also belongs there so it is just flipping the argument so you complete that argument to also infer that w plus u is contained inside v plus u and combining these two you have v plus u is equal to w plus u so the first one implying the second is done is there anything to prove in so far as the second one implying the third goes well that same so their intersection is either of them yeah can't be fine we are not interested in the trivial subspace right so obviously this is also done so this is done this is done the only thing that we then required to show is this this connection is what we have to establish right so how do we establish this if indeed as claim that this is a non empty therefore there is some common fellow sitting inside both of these right so there exists p such that p is equal to v plus u1 and p is equal to w plus u2 with u1 and u2 belonging to you right this is because p belongs to v plus u this is because p belongs to w plus u so what does this imply it is staring us in the face is it not so v plus u1 is equal to w plus u2 implying v minus w is equal to u2 minus u1 but what is u2 minus u1 if u1 and u2 belong to you then u2 minus u1 must also belong to this you which is really nothing but this connection so all of these three results or these three propositions are one and the same equivalent and now do you see why this implies that either two of those affine sets are either completely disjoint or they are identical to one another there is nothing in between right so this actually is good because this allows us to extend the notion of what is parallelism see parallelism is just something that we understand intuitively but here when you are talking about these affine sets they may not be from Euclidean space whatsoever but yet you can talk about them being parallel when there is nothing that is there in common between them then they are completely distinct and they are parallel and they are parallel precisely to a subspace an object that we recognize a subspace that is sitting inside v right and it is this result that allows us to do all the algebra that we want to with these quotient spaces because in the absence of this we would not know whether when we are defining these additions and scalar multiplications on objects in this vector space whether they even make sense or not right so that is the next object we shall now study which is how to define vector addition and scalar multiplication in the quotient space thereby rendering it with a vector space structure right any questions up until this point so far.